Regularized GRAPPA-based methods [1]} recover the full k-space \(\mathbb {Y}\)  from measurement \(\mathbb {Z}\) , then combine coils to map \(\mathbb {Y}\)  to \(\mathbb {X}\) . Since the coil sensitivity map is element-wise applied to each voxel through multiplication, each coil image is a modulated version of \(\mathbb {X}\)  and inherits a prior related to the prior on \(\mathbb {X}\) . For example, if \(\mathbb {X}\)  is \(K\) -sparse, then each coil image is at most \(K\) -sparse. The general optimization formulation of a regularized GRAPPA-based method is
\(\min _{\mathbb {Y}} \mathcal {D}(\mathbb {M}\circ \mathbb {Y},\mathbb {Z})+\lambda \mathcal {L}(\mathbb {Y})\) 
